Not in ohio if your married and your wife has a child thats not yours while you still being married to her the courts sees that child as yours hands down DNA test or not ive seen guys fight it and go to jail over child support of children who are not theirs.
This is a major problem, but it arises because of the very recent separation in the law between responsibility and authority.
Under the common law, children born to a married man are indeed his responsibility, without possibility of appeal and this is an issue for which actual (literal, genetic) paternity has no significance.
BUT....
Those that the common law made "children of his marriage" also were, by the common law, his chattels and in addition to being his sole responsibility they were his sole possessions.
The idea that some cheap whore could steal his children and make him pay for the favor was, and should still be, inconceiveable.